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Let me start with a brief reminder about cosmology:
1- The Universe is isotropic and homogeneous. 2- We assume General Relativity 3- The metric is the Friedmann-Lemaitre-Robertson-Walker for an expanding Universe.
22
2
2222
1dr
kr
drtadtds
22222 drdrdtds
The FLRW metric
222222 drdrtadtds
Minkowski metric you use in Special Relativity…
… but here the Universe is expanding. Spatial part multiplied by a scale factor solely function of time. Spatial coordinates are comoving. We can syncronize all the clocks in the universe at the same time t.a=1 today.
«Scale factor»
… the spatial part could also be non-Euclidean. k>0 means closed universe, k<0 means open universe. k is a constant.
«Curvature»
Friedmann’s Universe
22
2
2222
1dr
kr
drtadtds
Around 1922 Friedmann and, independently, Lemaitre (1927) independently proposed an expanding universe and therefore no need for a cosmological constant term. Einstein, at the beginning, rejected the idea of an expanding universe. In particular Einstein commented the idea as: “…. while mathematically correct it’s of no physical significance” While, according to Lemaitre, he was telling him: “Vos calculs sont corrects, mais votre physique est abominable”
Distance
Sp
eed
But an expanding universe his is exactly what was measured by Wirtz, Hubble and Humason …
We solve the system by introducing an equation of state of the form where w is usually a constant with time.
The Friedmann Equations
GTG 8
)(000
0)(00
00)(0
000
tP
tP
tP
t
T
2
2
2
000
000
000
0001
a
a
ag
2
2
3
8
a
kG
a
a
PG
a
a3
3
4
wP
We have two independent equations. The first one relates the expansion rate to the energy content. The second one the relates the acceleration to energy and pressure.
Einstein’s equation:
a
daw
d)1(3
The Continuity Equation Combining the first and second Friedmann equation we get:
We can consider 4 cases:
Matter 0P 0mw3
0
a
mm
4
0
a
rr
rrP
3
1
3
1rw
0
P 1w
Radiation
Cosmological Constant
iw
ii
a
13
0
We can integrate for a generic fluid with equation of state wi obtaining:
20 / akk 3/kkP 3/1kwCurvature
Some re-definitions
0
0
tta
aH
G
Hc
8
3 2
0
c
ii
0
The Hubble Constant:
The critical density (constant):
The density parameters (computed today):
Some re-definitions
Suppose to have i=1,..,n energy components with equation of state wi
ni
i
G
a
a
,..,1
2
3
8
ni
iiwG
a
a
,..,1
313
4
niw
i
iaHa
a
,..,1132
0
21
niw
ii
iaw
Ha
a
,..,1132
0
312
11
Curvature and the deceleration parameter
i
i1
i
ii
tt
wHa
aq 31
2
11
0
2
0
0
mk 1
2
0mq
Computing the Friedmann equations today, we have
Deceleration Parameter ! q0 > 0 universe is decelerating q0 < 0 universe is accelerating
In a model with curvature+matter+ cosmological constant:
If q0<0 then we need a cosmological constant. However we need to know the curvature to determine its precise value.
Measuring the deceleration parameter: luminosity distance.
In astronomy, given an object with known luminosity L we can measure its distance by measuring the radiative flux on earth:
24 r
Lf
Actually astronomers (since Hypparcos) use magnitudes
magnitude absolute
magnitudeapparent
ML
mf
pc
rMm
r
Lf
10log5
4102
Measuring the deceleration parameter: luminosity distance.
In cosmology, in an expanding universe, the luminosity distance is a function of the rate of expansion of the universe.
z
m
Lz
dzzcHzd
0
3
1
0'1
'1
...1
0 zcHzdL
...2
11 01
0
z
qzcHzdL
We can expand for small values of z. At first order we get the Hubble law: at second order we get deviations that depends on q0 !
252
11log5 01
0
z
qzcHMm
The Key Equation Measured is equivalent to the observed flux
Aestimated Equivalent to The intrinsic luminosity
Measured (spectra)
Derived Marginalized over
Objects with the same luminosity (or absolute magnitude) and at the same redshift will appear less luminous (or with greater apparent magnitude) if the universe is in accelerated expansion.
Decelerated Accelerated
Are Supernovae type Ia standard candles ?
During the course of the Calán/Tololo Supernova Survey, Mark M. Phillips discovered that the faster the supernova faded from maximum light, the fainter its peak luminosity was.
Riess et al., 1998 (submitted May 1998)
Perlmutter et al, 1999 Submitted Dec. 1998
Reasons to Believe: 1- Result confirmed by subsequent observations of high redshift SN-Ia
high-z SN-Ia discovered with HST
Reasons to Believe: 2- Result makes cosmology consistent with age of the Oldest objects and indication for a low density Universe.
In 1980 the most favoured model was the so-called cold dark matter model proposed by Peebles (1982) and others. This model is based on the Friedmann solution:
3
2
0
2
a
H
a
a
The CDM model is flat (the spatial part of the metric is euclidean) And the energy density of the universe is currently dominated by a matter component with solution:
0
2
3)(
3/4
3/1
3/2
3/2
0
ta
ta
tH
ta
Decelerated expansion.
Jim Peebles
Unfortunately, since the beginning of 1990, several problems for the CDM model started to emerge. In particular the age of the universe in the CDM model was too small if compared with the age of globular clusters:
GyrsHt 3.93
200
While ages for the globular clusters are in the range of 13-16 Billions of years.
Moreover, since the CDM model was flat, it was predicting an Energy density in the dark matter component equal to the critical energy density:
18
3 2
0 c
CDMMcCDM
G
H
Unfortunately observations from galaxy rotation curves and velocity dispersion in cluster of galaxies were suggesting a lower value:
3.0M
Bahcall et al, 1996
Finally the CDM model was predicting more galaxy clustering and clustering evolution than observed.
In 1995 Big Bang Model was nearly dead…
Reasons to Believe: 3- Other observables as CMB anisotropies are consistent with an accelerating universe.
two independent maps
Integrated Sachs-Wolfe map
Mostly large angular features
Early time map (z > 4)
Mostly from last scattering surface
Observed map is total of these, and has features of both (3 degree resolution)
Fosalba, Gaztanaga 2004
More than 5 ISW detections!
Mean redshift
Signal K Bias Catalog
Band
Reference
0.1 0.70 pm 0.32 1.1 2MASS, infrared
Afshordi et al. 2004
0.15 0.35 pm 0.17 1.0 APM, optical Scranton et al, 2004
0.3 0.26 pm 0.14 1.0 SDSS, optical
Fosalba et al. 2004
0.5 0.216 pm 0.1 1.8 SDSS
high z,
optical
Padmanabhan et al.
2004
0.9 0.04 pm 0.02 1-2 NVSS+
HEAO, Radio, X-Rays
Boughn & Crittenden 2004
Recent CMB data from ACT are sensitive to lensing and break geometrical degeneracy.
Sherwin et al, Phys.Rev.Lett.107:021302,2011
04.073.0
A model without cosmological constant is now ruled out at more than 18 sigma!
Zeldovich
“A new field of activity arises, namely the determination of ”
“The genie () has been let out of the bottle”
Already in 1968 Zeldovich noticed that the vacuum energy in particle physics could be a source for a cosmological constant.
The problem of the Cosmological Constant
1204
0
322 10p
M
vac MkdmkP
If we consider supersimmetry we go in the right direction but we are still 60 orders of magnitude away !
vac susy Msusy
4 1060
This anyway would lead to a great problem. The vacuum energy in particle physics is infinite. We may stop at Planck Scale but still we have a discrepancy of 120 orders of magnitude:
AM, Luca Pagano, Stefania Pandolfi arXiv:0706.131 Phys. Rev. D 76, 041301 (2007)
When did Cosmic acceleration start?
Alternatives to Lambda
- Scalar field: it can track the dominant component. Solves the «Why Now ?» problem. Mass of the field too small (ultralight and dark particle). - Modified Gravity: needs to «mimic» Lambda. Few models survive local constraints on deviation from GR. - Non homogeneous Universe: disomogeneities should decelerate the expansion not decelerate (Choudury theorem). No consistent picture present at the moment. - Anthropic principle. Landscape scenario….
COSMOLOGICAL COSTANT vs “Something else”
const
p
0
X (z) X (0)exp 3 dz'1 w(z')
1 z'0
z
pX w(z)X
X 0
Vs.
Perlmutter et al, 1998
WMAP Cosmological Parameters, Spergel et al., 2007
1.01w
Bayesian Model Selection
E P D | H P D | ,H P ,H
Likelihood Prior
Jeffrey(1961):
1 ln(E) 2.5 Substantial
2.5 ln(E) 5 Strong
5 ln(E) Decisive
Current cosmological data are in agreement with more complicated Dark energy parametrizations, but do we need more parameters ? More complicated models should give better fits to the data. In model selection we have to pay the larger number of parameters (see e.g. Mukherjee et al., 2006):
Evidence
P. Serra, A. Heavens, A. Melchiorri Astro-ph/0701338 MNRAS, 379, 1,169 2007
More Parameters
Current data: “Substantial” Evidence for a cosmological constant…
Dark Energy- a recent analysis for w(z)
● We sample w(z) in 5 redshift bins up to z=1:
● We use CMB (WMAP5,QUAD, ACBAR) data,
BAO DR7, Weak Lensing from CFHFTLS, ISW
data, Supernovae from UNION and
CONSTITUTION datasets.
Serra, Cooray, Holtz, Melchiorri, Pandolfi, Sarkar, Phys.Rev.D80:121302,2009
Serra, Cooray, Holtz, Melchiorri, Pandolfi, Sarkar, Phys.Rev.D80:121302,2009
CMB+WL+BAO+UNION+ISW
Serra, Cooray, Holtz, Melchiorri, Pandolfi, Sarkar, Phys.Rev.D80:121302,2009
No evidence from current data for deviations from a cosmological constant
http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48983#
http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48983#
For almost a century, the Universe has been known to be expanding as a consequence of the Big Bang about 14 billion years ago. However, the discovery that this expansion is accelerating is astounding. If the expansion will continue to speed up the Universe will end in ice. The acceleration is thought to be driven by dark energy, but what that dark energy is remains an enigma - perhaps the greatest in physics today. What is known is that dark energy constitutes about three quarters of the Universe. Therefore the findings of the 2011 Nobel Laureates in Physics have helped to unveil a Universe that to a large extent is unknown to science. And everything is possible again.
